Strictly Hermitian Positive Definite Functions
نویسنده
چکیده
Let H be any complex inner product space with inner product < ·, · >. We say that f : | C → | C is Hermitian positive definite on H if the matrix ( f(< z,z >) )n r,s=1 (∗) is Hermitian positive definite for all choice of z, . . . ,z in H, all n. It is strictly Hermitian positive definite if the matrix (∗) is also non-singular for any choice of distinct z, . . . ,z in H. In this article we prove that if dimH ≥ 3, then f is Hermitian positive definite on H if and only if f(z) = ∞ ∑ k,`=0 bk,`z k z ` (∗∗) where bk,` ≥ 0, all k, ` in ZZ+, and the series converges for all z in | C. We also prove that f of the form (∗∗) is strictly Hermitian positive definite on any H if and only if the set J = {(k, `) : bk,` > 0} is such that (0, 0) ∈ J , and every arithmetic sequence in ZZ intersects the values {k− ` : (k, `) ∈ J} an infinite number of times. §
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